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In this paper, the self-propelled movement on gyrotactic swimming microorganisms into this generalized slip flow by nanoliquid over the stretching cylinder with strong magnetic field is discussed. Constant wall temperature was pretended as well as the Nield conditions of boundary. The intuitive non-Newtonian particulate suspension was included into applying Casson fluid by the base liquid and nanoparticles. This formation on the bio-mathematical model gives the boundary value problem by the nonlinear partial differential equations. Primly, modeled numerical system was converted to nondimensional against this help on acceptable scaling variables and the bvp4c technique was used to acquire the mathematical outcomes on the governing system. This graphical description by significant parameters and their physical performance was widely studied. The Prandtl number has the maximum contribution (112.595%) along the selected physical parameters, whereas the Brownian motion has the least (0.00165%) heat transfer rate. Anyhow, Casson fluid was established for much helpful suspension of this method on fabrication and coatings, etc. Therefore, this magnetic field performs like the resistive force of that fluid motion, and higher energy was enlarged into the structure exhibiting strong thermal radiation.

This article analyzes the typical Langevin problem, i.e. the dynamics of a charged particle under the circularly successive influence of several simple impulse functions moving in a double-well potential and a time-dependent magnetic field. Using the stroboscopic sampling, by selecting an appropriate magnetic intensity and time interval, we reduce the Langevin equation to a class complex mapping system. Through an experimental mathematical method, the authors study the structures of generalized M–J (Mandelbrot–Julia) sets generated by the complex mapping system, and expatiate the theory of Brownian movement. The authors find that:.

(1) This paper extends Shirriff constructed M sets by combining two simple complex mappings;.

(2) The fractal structure of the generalized M–J sets may visually depict the rule of Brownian movement, and the infinite overlapping embedment self-similar structure reflects the complexity of Brownian movement;.

(3) Whether the selected time interval is significant or not determines the continuity of the fractal structure for the generalized M–J sets;.

(4) The changing rule of particle velocity depends on the different choices of the principal range of phase angle;.

(5) If we change the choices of the magnetic intensity and time interval, for example, we choose a randomly fluctuating magnetic field, the generalized J sets may emerge the interior-filling structure feature, i.e. "explosion" phenomena appear in the closure of the unstable periodic orbits of the particle in the velocity space.

In recent times, the interaction of nanoparticles has significantly enhanced the thermal association of heat transport. This phenomenon plays a crucial role in hydraulic systems, particularly in the context of lubrication and its associated consequences on mass and heat transport. Current studies have focused on investigating the thermal effects of a third-order nanofluid on a lubricated stretched surface near an analytical stagnation point. The lubrication process involves the use of a thin, adjustable coating of lubricant fluid. To analyze this complex system, we employ the Buongiorno model and explore thermophoresis and the Brownian motion phenomenon. For deriving analytical results of updated boundary layer ordinary differential equations, we rely on the dependable and effective hybrid homotopy analysis method (HHAM). To exhibit the effectiveness of our study, we provide a numerical comparison. Based on theoretical flow assumptions, we establish a range of flow parameters. In the presence of lubrication, we physically examine how these parameters affect temperatures, velocities, concentration, and other relevant quantities of thermal interest. These new findings have practical applications in polymer production, heat transmission, and hydraulic systems.